Relations of loop partial amplitudes in gauge theory by Unitarity cut method

It is well known that under the color-decomposition, one-loop amplitude of gluons contains partial amplitudes of single and double trace structures, and particularly all partial amplitudes of double trace structure can be expressed as a linear combination of partial amplitudes of single trace structure. Using unitarity cut method, we prove that this result is the natural consequence of tree-level Kleiss-Kuijf relation. Generalizing the unitarity cut method to two-loop (triple cut in this case), we show that, unlike the one-loop case, partial amplitudes of double and triple trace structures cannot be expressed as a linear combination of partial amplitudes of leading-color single trace structure. For partial amplitudes of subleading-color single trace structure, we have shown a very non-trivial Kleiss-Kuijf relation for six and seven-point amplitudes, which is one new result of our paper and cannot be obtained by U(1)-decoupling method. Mysteriously, when we consider the case of eight points, Kleiss-Kuijf relation must be modified for subleading-color single trace partial amplitudes.